In this paper ideal matrices with respect to ideals in the maximal order of an algebraic number field are connected with the different of the field and with group matrices in the case of normal fields whose maximal order has a normal basis

Taussky, Olga

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PACIFIC JOURNAL OF MATHEMATICS
Vol. 118, No. 2,1985
IDEAL MATRICES. Ill
OLGA TAUSSKY
Dedicated to the memory of Ernst G. Straus
In this paper ideal matrices with respect to ideals in the maximal
order of an algebraic number field are connected with the different of the
field and with group matrices in the case of normal fields whose maximal
order has a normal basis.
1. Let F be an algebraic number field of degree n over Q. Let £) be
the maximal order of F and α an ideal in £). Let ωl9... ,ωw be a Z-basis of
£> and al9... ,an a Z-basis of α. Then there exists an integral n X n άx
A such that
(1)
A is called an ideal matrix for α. The ideal defines a representation
module for D. A change of bases replaces A by UAV where Ϊ7, F are
unimodular n X n Z-matrices and conversely, to any U9 V there exist
bases for £) and α such that UAV is an ideal matrix for α.
2. THEOREM 1. LeJ (y) denote the family of embeddings of F in the
complex numbers. Then, suppressing the superscript (1),
(2)
0) (2)
Λ < 2 )
x<">
Hence we have
THEOREM 2. A = v-l
Both theorems have trivial proofs. Let a be any integral generator of
F. Then there exists an integral matrix X such that detίωp^det X =
± different α. Further, (ωPXωp*)' is the so-called discriminant or trace
matrix of £) where ' stands for the transpose and the determinant of the
above matrix product is the discriminant of the maximal order.
599
600 OLGA TAUSSKY
REMARK. In an earlier paper by this author and M. Newman, Comm.
Pure Appl. Math. 9 (1956), 85-91, the matrix (ωp>) was called the
discriminant matrix. This term is at present being used by other authors
for the product of the matrix and its transpose.
3. The case of a normal field F.
THEOREM 3. The matrix {ω\j)) is a group matrix in the sense of [1]
provided that the ω's are a normal basis for €).
Proof. For a suitable ordering of the elements σ
x
 = I,σ2y...,σn,of the
Galois group G of F the following equation can be made to hold
r τ - 1(3) ωp> = ωf
If α too has a normal basis then A is the product of two group matrices.
4. Ideal matrices for products of ideals and for the conjugate ideal.
THEOREM 4. (The so-called A UB theorem, see [Taussky 3] Theorem 1):
Let A, B be ideal matrices corresponding to the ideals Q, 6 i« fl. Then there
exists an integral unimodular matrix U such that A UB is an ideal matrix for
the product αb.
THEOREM 5. The matrix equation (2) for the ideal α ( y ) is obtained
immediately since the matrix (αp*) consists of columns of conjugates.
5. The ideal matrix and Stickelberger theory for F a field generated by
a root of unity ξ
m9 see e.g. [Ireland and Rosen, [4] page 246]. In the case
under consideration every prime ideal not containing m, raised to a
certain symbolic power coming from the integral group ring of the Galois
group of F is a principal ideal. From the tools developed in 4 the
corresponding idea can be applied to A.
REFERENCES
[1] O. Taussky, A note on group matrices, Proc. Amer. Math.. Soα, 6 (1955), 984-986.
[2] , Ideal matrices J, Archiv d. Math., (13) (1962), 275-282.
[3] , Ideal matrices II, Math. Ann., 150 (1963), 218-225.
[4] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory,
Springer 1980.
[5] H. Jacobinski, ΪJber die Ήauptordnung eines K'όrpers als Gruppenmodul', J. f. d. reine
und angew. Math., 213 (1964), 151-164.
IDEAL MATRICES. Ill 601
[6] D. Maurer, Stίckelberger's criterion, Galois algebras and tame ramifications, J. Pure
and Appl. Algebra, 33 (1984), 281-293.
[7] L. R. McCulloh, A Stickelberger condition on Galois module structure for Kummer
extensions of prime degree, Algebraic number fields, Editor A. Frδhlich, Academic
Press, 1977.
[8] S. Ullom, Integral representations afforded by ambiguous ideals in some abelian exten-
sions, J. Number Theory, 6 (1974), 32-49.
[9] H. Yokoi, On the ring of integers in an algebraic number field as a representation
module of Galois group, Nagoya Math. J., 16 (1960), 83-90.
[10] , A cohomological investigation of the discriminant of a normal algebraic number
field, Nagoya Math. J., 27 (1966), 207-211.
Received June 20, 1984.
CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CA 91125



Ideal matrices. III.

A Classical Introduction to Modern Number Theory,

A cohomological investigation of the discriminant of a normal algebraic number field,

A cohomological investigation of the discriminant of a normal algebraic number field, Nagoya Math. J., 27 (1966), 207-211. Received June 20, 1984. CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CA 91125

A note on group matrices,

A Stickelberger condition on Galois module structure for Kummer extensions of prime degree, Algebraic number fields, Editor A. Frδhlich,

D. Maurer, Stίckelberger's criterion, Galois algebras and tame ramifications, J. Pure and Appl. Algebra, 33 (1984), 281-293.

H. Jacobinski, ΪJber die Ήauptordnung eines K'όrpers als Gruppenmodul', J. f. d. reine und angew. Math., 213 (1964), 151-164. IDEAL MATRICES. Ill 601

H. Yokoi, On the ring of integers in an algebraic number field as a representation module of Galois group, Nagoya Math. J., 16 (1960), 83-90.

Ideal matrices II, Math. Ann., 150 (1963), 218-225.

Ideal matrices J, Archiv d. Math., (13) (1962), 275-282.

Integral representations afforded by ambiguous ideals in some abelian extensions,

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer 1980.

L. R. McCulloh, A Stickelberger condition on Galois module structure for Kummer extensions of prime degree, Algebraic number fields, Editor A. Frδhlich, Academic Press, 1977.

O. Taussky, A note on group matrices, Proc. Amer. Math.. Soα, 6 (1955), 984-986.

On the ring of integers in an algebraic number field as a representation module of Galois group,

S. Ullom, Integral representations afforded by ambiguous ideals in some abelian extensions, J. Number Theory, 6 (1974), 32-49.

Stίckelberger's criterion, Galois algebras and tame ramifications,

ΪJber die Ήauptordnung eines K'όrpers als Gruppenmodul',

https://authors.library.caltech.edu/836/1/TAUpjm85.pdf

Ideal matrices. III.

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